Solving force problems step by step

Why do force problems feel hard?

Many force problems mix two chapters: forces, and the equations of motion. A plan with four steps keeps them apart.

1. Draw the object and every force on it, with arrows.
2. Net force: add the forces with their directions.
3. The law: $a = \dfrac{F}{m}$.
4. Motion: use $v = u + at$, $s = ut + \tfrac12 at^2$ or $v^2 = u^2 + 2as$.

Some problems run the plan backwards: from the motion to the acceleration, then to the force.

You should find that with the push no bigger than the friction, nothing happens: the friction matches the push and the speed stays zero.

Watch out. Do not put the mass into the equations of motion, or a speed into $F = ma$. Step 3 joins the two halves.

A worked example

A $50\ \text{kg}$ crate at rest is pushed with $300\ \text{N}$ against $150\ \text{N}$ of friction for $4\ \text{s}$. Find its speed and how far it goes.

1. Draw. Push right $300\ \text{N}$, friction left $150\ \text{N}$.
2. Net force. $300\ \text{N} - 150\ \text{N} = 150\ \text{N}$.
3. The law. $a = \dfrac{150\ \text{N}}{50\ \text{kg}} = 3\ \text{m/s}^2$.
4. Motion. $v = 3\ \text{m/s}^2 \times 4\ \text{s} = 12\ \text{m/s}$ and $s = \tfrac12 \times 3\ \text{m/s}^2 \times (4\ \text{s})^2 = 24\ \text{m}$.

Practise with new numbers

Try it in Python

The program follows the four steps.

Remember these

• Draw, net force, law, motion.
• Step 3, $a = F/m$, links forces to motion.
• The plan works backwards too: motion, then $a$, then force.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A $20\ \text{kg}$ cart has $90\ \text{N}$ forward and $30\ \text{N}$ backward on it. What is its acceleration?

2. Starting from rest with that acceleration, how fast is it after $5\ \text{s}$?

3. A $600\ \text{kg}$ car slows from $10\ \text{m/s}$ to rest in $5\ \text{s}$. What is the braking force?

4. Which step would you skip if you were asked only for the acceleration?